Redundancy allocation with component costs

(s1, s2, s3, s4) sequence initial reliability final reliability (5,5,5,5) 2-3-1-2-3-2-3-1-3-2-1-4 0.7733 0.9742 (5,5,5,4) 2-3-1-2-3-4-2-3-1-4-3-2 0.7485 0.9595 (5,5,5,3) 2-3-4-1-2-4-3-4-2-4-3-1 0.6867 0.9295 (5,5,5,2) 4-2-4-3-4-1-4-2-4-3-4-4 0.5630 0.8556 (5,5,5,1) 4-4-4-2-4-4-3-4-1-4-4-2 0.3464 0.6460

Table 2.8: Optimal allocation sequences of 12 components

For the zero-failure case, so with si _{= 5 for all} _i _{= 1, . . . ,}_{4, this example was}

also presented by Coolen-Schrijneret al.[29] who showed that, due to the fact that subsystem 4, a 1-out-of-4 (parallel) system, has the largest built-in redundancy, the first extra component added to this subsystem is actually only the 12th in the optimal allocation sequence. This sequence of the first 12 extra components is presented again in Table 2.8, together with corresponding sequences for situations with one or more components of type 4 failing in the test, while no other components failed. This clearly illustrates that, for an increasing number of failed components of a particular type in the test, one allocates extra components to the corresponding subsystem earlier in the optimal sequence. For the last case, with only 1 out of 5 components of type 4 functioning successfully in the test, one clearly adds a large number of extra components to subsystem 4, but the effect of reduced component reliability still causes the final reliability to be substantially smaller than for the other test results and optimal allocation sequences reported.

2.6 Redundancy allocation with component costs

This section presents the inclusion of different costs per component of the different types. The results in the previous section determine how to optimally allocate additional components for redundancy for any criterion in the case where the cost of components is irrelevant, or where they are the same for all components. If the costs of additional components differ per subsystem, and one aims to maximize system reliability under budget constraints, then the redundancy allocation problem

2.6. Redundancy allocation with component costs 30

becomes more complex. This problem can be formulated as follows:

Let ci _{be the cost to add one extra component to subsystem} _{i. Then the total}

cost of these additional components being added to the whole system is:

C(j) = C(j1, . . . , jL) =

∑

i=1

ciji

Obtaining optimal system reliability with a ﬁxed budget B means that we need to add J additional components to the whole system (J =∑L_i₌₁ji) in order to

maximize L ∏ i=1 ji₋₁ ∏ li₌₀ ρ(i, li)

subject to the restriction

∑

i=1

ciji 6B ji ≥0 ∀i= 1, . . . , L

This goal function can be replaced by: maximize∑L_i₌₁∑j_lii₌₀−1ln(ρ(i, li)). This prob-

lem is close in nature to the well-known knapsack problems in discrete optimisa- tion [43]. The knapsack problem is a problem of how to choose items to maximize their total value under a constraint of maximal weight. Let us assume that we can choose from items 1, . . . , nwith weights a1_{, a}2_{, . . . , a}n_{and proﬁts}_p1_{, p}2_{, . . . , p}n_{. The}

capacity of the knapsack K ∈ N is also given. The task is now to select a subset of the items so that its total weight does not exceed K and its proﬁt is maximized among those subsets. The integer program formulation of the knapsack problem is the following. For all i= 1, . . . , n we have a variablexi _{∈ {}_0,₁_}_,

maximize ∑pixi subject to

∑

aixi 6K

There are diﬀerent versions of the knapsack problem [43], for example the single knapsack problem is the case where one container (or knapsack) must be ﬁlled with an optimal subset of items. If more than one container is available, the multiple knapsack problem will be considered. Also, according to the number of copies allo- cated of each item one can distinguish between the unbounded knapsack problem,

2.6. Redundancy allocation with component costs 31 i ki mi ci 1 3 4 5 2 2 3 4 3 4 6 3 4 2 4 6

Table 2.9: Subsystemi: ki_-out-of-mi

which places no bound on the number of each item, and the bounded knapsack problem, which restricts the number of each item to a maximum value. The typical formulation in practice is the 0-1 knapsack problem, where only one copy of each item is available.

The system considered in this chapter consists of series conﬁgurations ofLinde- pendentki_-out-of-mi _{subsystems. For each}_i_{= 1, . . . , L,}_{ρ(i, j}i_{) is strictly decreasing}

in ji, but ci is assumed to be ﬁxed. It means that the extra components to be al- located with cost (weight) ci _{and utility (value) ln}_{ρ(i, l}i_{) are not the same. This}

allocation problem is considered as a 0-1 knapsack problem, which can be solved by basic dynamic programming.

Example 2.5

The redundancy allocation under fixed budget B using a knapsack problem formulation is illustrated via a basic system consisting initially of four independent ki-out-of-mi subsystems in series configuration, with the values ki, mi and ci as given in Table 2.9. Several scenarios of allocation of additional components, under different budgets, will be illustrated for this system. Throughout this example, we assume that 5 components of each type were tested, so ni _{= 5 for} _i _{= 1, . . . ,}_{4. To}

concentrate on the eﬀect of the budgetB, we assume zero-failure testing, so si = 5 fori= 1, . . . ,4.

In document Nonparametric Predictive Inference for System Reliability (Page 38-41)